A convex analytic approach to system identification

Abstract

This paper introduces a new concept for system identification in order to account for random and nonrandom(deterministic/set-membership) uncertainties. While, random/stochastic models are natural for modeling measurement errors, nonrandom uncertainties are well-suited for modeling parametric and nonparametric components. The new concept introduced is distinct from earlier concepts in many respects. First, inspired by the concept of uniform convergence of empirical means developed in machine learning theory, we seek a stronger notion of convergence in that the objective is to obtain probabilistic uniform convergence of model estimates to the minimum possible radius of uncertainty. Second, the formulation lends itself to convex analysis leading to description of optimal algorithms, which turn out to be well-known instrument-variable methods for many of the problems. Third, we characterize conditions on inputs in terms of second-order sample path properties required to achieve the minimum radius of uncertainty. Finally, we present fundamental bounds and optimal algorithms for system identification for a wide variety of standard as well as nonstandard problems that include special structures such as unmodeled dynamics, positive real conditions, bounded sets and linear fractional maps.